Lemma 24.31.1. In the situation above the differential graded $\mathcal{O}$-algebra

$\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i$

has the following property: for any morphism $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ of ringed topoi, the pullback $f^*\mathcal{A}$ is flat as a graded $\mathcal{O}'$-module and is K-flat as a complex of $\mathcal{O}'$-modules.

Proof. Observe that $f^*\mathcal{A} = \mathop{\mathrm{colim}}\nolimits f^*\mathcal{A}_ i$ and that

$f^*\mathcal{A}_ i = \mathcal{O}'\langle f^{-1}\mathcal{S}_0 \amalg \ldots \amalg f^{-1}\mathcal{S}_ i\rangle$

with differential given by the inductive procedure above using $f^{-1}\delta _{i + 1}$. Thus it suffices to prove that $\mathcal{A}$ is flat as a graded $\mathcal{O}$-module and is K-flat as a complex of $\mathcal{O}$-modules. For this it suffices to prove that each $\mathcal{A}_ i$ is flat as a graded $\mathcal{O}$-module and is K-flat as a complex of $\mathcal{O}$-modules, compare with Lemma 24.23.3.

For $i \geq 1$ write $\mathcal{S} = \mathcal{S}_0 \amalg \ldots \amalg \mathcal{S}_ i$ so that we have $\mathcal{A}_ i = \mathcal{O}\langle \mathcal{S} \rangle$ as a graded $\mathcal{O}$-algebra. We are going to construct a filtration of this algebra by differential graded $\mathcal{O}$-submodules.

Set $W = \mathbf{Z}_{\geq 0}^{i + 1}$ considered with lexicographical ordering. Namely, given $w = (w_0, \ldots w_ i)$ and $w' = (w'_0, \ldots , w'_ i)$ in $W$ we say

$w > w' \Leftrightarrow \exists j,\ 0 \leq j \leq i : w_ i = w'_ i,\ w_{i - 1} = w'_{i - 1},\ \ldots , \ w_{j + 1} = w'_{j + 1},\ w_ j > w'_ j$

and so on. Suppose given a section $s = s_1 \cdot \ldots \cdot s_ r$ of $\mathcal{S} \times \ldots \times \mathcal{S}$ over $U$. We say that the weight of $s$ is defined if we have $s_ a \in \mathcal{S}_{j_ a}(U)$ for a unique $0 \leq j_ a \leq i$. In this case we define the weight

$w(s) = (w_0(s), \ldots , w_ i(s)) \in W,\quad w_ j(s) = |\{ a \mid j_ a = j\} |$

The weight of any section of $\mathcal{S} \times \ldots \times \mathcal{S}$ is defined locally. The reader checks easily that we obtain a disjoint union decompostion

$\mathcal{S} \times \ldots \times \mathcal{S} = \coprod \nolimits _{w \in W} \left( \mathcal{S} \times \ldots \times \mathcal{S}\right)_ w$

into the subsheaves of sections of a given weight. Of course only $w \in W$ with $\sum _{0 \leq j \leq i} w_ j = r$ show up for a given $r$. We correspondingly obtain a decomposition

$\mathcal{A}_ i = \mathcal{O} \oplus \bigoplus \nolimits _{r \geq 1} \bigoplus \nolimits _{w \in W} \mathcal{O}[\left(\mathcal{S} \times \ldots \times \mathcal{S}\right)_ w]$

The rest of the proof relies on the following trivial observation: given $r$, $w$ and local section $s = s_1 \cdot \ldots \cdot s_ r$ of $\left(\mathcal{S} \times \ldots \times \mathcal{S}\right)_ w$ we have

$\text{d}(s) \text{ is a local section of } \mathcal{O} \oplus \bigoplus \nolimits _{r' \geq 1} \bigoplus \nolimits _{w' \in W,\ w' < w} \mathcal{O}[\left(\mathcal{S} \times \ldots \times \mathcal{S}\right)_{w'}]$

The reason is that in each of the expressions

$(-1)^{\deg (s_1) + \ldots + \deg (s_{a - 1})} s_1 \cdot \ldots s_{a - 1} \cdot \delta (s_ a) \cdot s_{a + 1} \cdot \ldots \cdot s_ r$

whose sum give the element $\text{d}(s)$ the element $\delta (s_ a)$ is locally a $\mathcal{O}$-linear combination of elements $s'_1 \cdot \ldots \cdot s'_{r'}$ with $s'_{a'}$ in $\mathcal{S}_{j'_ a}$ for some $0 \leq j'_{a'} < j_ a$ where $j_ a$ is such that $s_ a$ is section of $\mathcal{S}_{j_ a}$.

What this means is the following. Suppose for $w \in W$ we set

$F_ w \mathcal{A}_ i = \mathcal{O} \oplus \bigoplus \nolimits _{r \geq 1} \bigoplus \nolimits _{w' \in W,\ w' \leq w} \mathcal{O}[\left(\mathcal{S} \times \ldots \times \mathcal{S}\right)_{w'}]$

By the observation above this is a differential graded $\mathcal{O}$-submodule. We get admissible short exact sequences

$0 \to \mathop{\mathrm{colim}}\nolimits _{w' < w} F_{w'}\mathcal{A}_ i \to F_ w\mathcal{A}_ i \to \bigoplus \nolimits _{r \geq 1} \mathcal{O}[\left(\mathcal{S} \times \ldots \times \mathcal{S}\right)_ w] \to 0$

of differential graded $\mathcal{A}$-modules where the differential on the right hand side is zero.

Now we finish the proof by transfinite induction over the ordered set $W$. The differential graded complex $F_0\mathcal{A}_0$ is the summand $\mathcal{O}$ and this is K-flat and graded flat. For $w \in W$ if the result is true for $F_{w'}\mathcal{A}_ i$ for $w' < w$, then by Lemmas 24.23.3, 24.23.2, and 24.23.6 we obtain the result for $w$. Finally, we have $\mathcal{A}_ i = \mathop{\mathrm{colim}}\nolimits _{w \in W} F_ w\mathcal{A}_ i$ and we conclude. $\square$

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